Afamily lines up for a photograph. in each of the following situations, how many ways are there for the family to line up so that the mother is next to at least one of her daughters? (a) the family consists of two parents, two daughters and two sons. solution call the daughters a and b. the number of ways the family can line up with the mother next to a is 2-5! . the number of ways the family can line up with the mother next to b is 2.5! to determine the number of ways in which the family can line up in which the mother is next to both a and b, first decide whether a is to the left and b is to the right of the mother or whether a is to the right and b is to the left of the mother (2 choices). once this decision has been made, stick the daughters to either side of the mother. the number of items to permute is now four: father, two sons, and the daughter-mother-daughter unit. there are 4! ways to permute the four people/groups. thus, there are 2-4! ways to line up the family so that the mother is next to both daughters. use the inclusion-exclusion principle to determine the number of ways to line up the family so that the mother is next to one or both of her daughters: 2.5! + 2.5! - 2.4! 45! -2.4! (b) the family consists of two parents, three daughters and four sons.